Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modelling questions

Abstract

This paper deals with three fundamental modelling questions for the Darcy and Brinkman equations for flow in a porous medium. It is first shown that in the Dirichlet initial — boundary value problem for the Brinkman equations the solution depends continuously on the viscous coefficient. Then L2 convergence of the solution of this problem to the solution of an analogous problem for the Darcy equations is established. Finally, it is proved that for flow in a domain occupied by a viscous fluid in contact with a porous solid, the solution depends continuously on a coefficient in the interface boundary condition. The continuous dependence holds for Stokes flow in the fluid, and the analogous Navier-Stokes situation is discussed.